本文研究与纯断局部鞅相关的指数型集中不等式;通过构造一个新颖的指数鞅并利用停止定理,建立关于纯断局部鞅的一般化集中不等式.作为推论,本文在不同条件下得到了一系列指数型不等式,包括纯断局部鞅的Bernstein型不等式和de la Pena不...本文研究与纯断局部鞅相关的指数型集中不等式;通过构造一个新颖的指数鞅并利用停止定理,建立关于纯断局部鞅的一般化集中不等式.作为推论,本文在不同条件下得到了一系列指数型不等式,包括纯断局部鞅的Bernstein型不等式和de la Pena不等式等.此外,本文考虑连续时间矩阵值局部鞅,通过构造迹指数上鞅,建立矩阵值纯断局部鞅算子范数的集中不等式,改进了Bacry等(2018)的结果.展开更多
Let Xn be a standard real symmetric （complex Hermitian） Wigner matrix, Y1,Y2,...,Yn a sequence of independent real random variables independent of Xn. Consider the deformed Wigner matrix Hn,a = n^（-1/2）Xn ＋ n^（-...Let Xn be a standard real symmetric （complex Hermitian） Wigner matrix, Y1,Y2,...,Yn a sequence of independent real random variables independent of Xn. Consider the deformed Wigner matrix Hn,a = n^（-1/2）Xn ＋ n^（-a/2）diag（y1,...,yn）, where 0 〈 a 〈 1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform mn,a（z） converges in probability to the corresponding Stieltjes transform rn（z）. In this paper, we shall give the asymptotic estimate for the expectation Emn,a （z） and variance Var（mn,a （z））, and establish the central limit theorem for linear statistics with sufficiently regular test function. A basic tool in the study is Stein＇s equation and its generalization which naturally leads to a certain recursive equation.展开更多
文摘本文研究与纯断局部鞅相关的指数型集中不等式;通过构造一个新颖的指数鞅并利用停止定理,建立关于纯断局部鞅的一般化集中不等式.作为推论,本文在不同条件下得到了一系列指数型不等式,包括纯断局部鞅的Bernstein型不等式和de la Pena不等式等.此外,本文考虑连续时间矩阵值局部鞅,通过构造迹指数上鞅,建立矩阵值纯断局部鞅算子范数的集中不等式,改进了Bacry等(2018)的结果.
基金Much of the work was done when the author was visiting the Department of Mathematics, Harvard University, under the project from the Y. C. Tang Disciplinary Development Fund, Zhejiang University. The author thanks Professor H. T. Yau and Professor S. T. Yau for their hospitality during the visit. The referees＇ careful reading helps to improve the presentation of the paper. This work was Partially supported by the National Natural Science Foundation of China （Grant No. 11071213）, the Natural Science Foundation of Zhejiang Province （No. R6090034）, and the Doctoral Program ~and of Ministry of Education （No. J20110031）.
文摘Let Xn be a standard real symmetric （complex Hermitian） Wigner matrix, Y1,Y2,...,Yn a sequence of independent real random variables independent of Xn. Consider the deformed Wigner matrix Hn,a = n^（-1/2）Xn ＋ n^（-a/2）diag（y1,...,yn）, where 0 〈 a 〈 1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform mn,a（z） converges in probability to the corresponding Stieltjes transform rn（z）. In this paper, we shall give the asymptotic estimate for the expectation Emn,a （z） and variance Var（mn,a （z））, and establish the central limit theorem for linear statistics with sufficiently regular test function. A basic tool in the study is Stein＇s equation and its generalization which naturally leads to a certain recursive equation.
